That wouldn't happen with me. I occasionally walk friends and neighbors dogs but would never take them to an off leash dog park. There's way to much that can go wrong, I'm not likely to have the dogs under verbal control, and I do not know them well enough to be able to anticipate what dogs they may not mix well with. I stick to on leash walks in the neighborhood the dogs live in.
In your hypothetical situation I'd give the other owner my contact info and be willing to pay the vet bills and take full responsibility. It might be another persons dog but I'm the one that brought the dog to the dog park and I'm the one who should have been in control of the dog. The fault lies with me, not the dog's owner and not the dog.